Physics o P Black Holes Patrick Bruskiewich June, 2002 t 970 CHAPTER unitization Figure 1 How the geometry of space is warped by a spherical (nonro- tating) black hole. The radial distance from any point to the center of the black hole can be measured in two ways. One way is to measure ihe circum- ference of a circle passing through the point and cen- tered on the hole, and divide by 2tt. A second way is to measure the distance from the point to the center along a radial line on the curved sur- face. These two measures will be nearly equal far from the hole where space is "flat," but highly discrepant near the black-hole horizon where the funnel narrows to a ver- tical tube. Black Holes Alan P. Lightman Cornell University Imagine a region of space where attractive gravitational force is so intense that light rays venturing too near can be bent into circular orbit -a region from which no matter, radiation, or communication of any kind can ever escape. Such is a black hole, one of the most exciting creatures of theoretical physics and possibly the most bizarre object of space. Although they were implicitly predicted by Einstein's 1915 theory of grav- ity, general relativity, black holes were first theoretically "discovered" by Op- penheimer and Snyder in 1939. Because of their highly nonintuitive proper- ties, however, black holes were not taken seriously by most physicists and astronomers until the mid-1960s. We are now on the verge of confirming the discovery in space of the first black hole. The concept of a black hole stretches our ideas of time and space to their limits. The surface of a black hole, called its horizon, is a closed boundary within which the escape velocity exceeds the velocity of light. The prediction of such a surface for sufficiently compact bodies can be made just on the basis of Newton's theory of gravity together with special relativity: the escape veloc- ity of a particle launched from the surface of a spherical mass M of radius R is v e = V2GM/R (see Chapter 9). When MIR satisfies 2GM/R > c 2 , v e exceeds the speed of light and no particle or photon may escape, as required by special rel- ativity. As a remarkable result, the interior of a black hole is causally discon- nected from the rest of the universe; no physical process occurring inside the horizon can communicate its existence or effects to the outside. For a spherical black hole of mass M, the horizon is a sphere of circumference equal to 2ir times the Schwarzschild radius of the hole Re, where Re = 2GM/c 2 (exact nu- merical coincidence of this radius with the newtonian analog above is ac- cidental). A black hole with a mass equal to that of our sun would have a Schwarzschild radius of 2.95 km. According to general relativity, time and space are warped by the gravita- tional field of massive bodies, with the warping most severe near a biack hole. Gravity affects all physical systems in a universal way so that all clocks (whether transitions of an ammonia molecule or heartbeats of a human being) and all rulers would indicate that time is slowed down and space stretched out near a black hole (see Figure 1). Alternatively, one can describe the effects of the hole's gravitational field on a local measurement of time and space in-' tervals as an acceleration of the local Lorentz frame (in which special relativity is valid) with respect to other local Lorentz frames at different locations. Black holes are formed when massive stars undergo total gravitational collapse. While emitting heat and light into space, stars balance themselves against their own gravity with the outward thermal pressure force generated by heat from nuclear reactions in their deep interiors. But every star must die. When its nuclear fuel is exhausted, a star will contract. If its mass is Jess fen about three times the mass of our sun, the shrinking star will stabilize at a smaller size when the inward pull of gravity cannot force the star's constituent particles any closer together. Such a star will spend eternity as a white dwarf or neutron star. But if the star's mass exceeds about 3 solar masses, theory in- dicates that no amount of outward pressure force can stave off the overwhelm- ing crush of gravity and the star will plunge in upon itself, disappearing for- ever from sight and giving birth to a black hole. Recent mathematical proofs, using Einstein's general relativity theory, dem- onstrate that a black hole is one of the simplest objects of nature and can be completely characterized by only three quantities: its mass, angular momen- tum, and net electrical charge. Other than these three quantities, all informa- tion of the progenitor star, e.g., whether it was composed of particles or antiparticles, whether it was pancake-shaped or spherical, is lost via gravita- .: ' ■" : ' ' : REVIEW 97! . ■ ■ . tonal and electromagnetic waves shortly after formation of the hole. Rotating sur o, JTk C f h °u S (n ° nrotat!n g hol « ^ "lied SchwarzschM holes), surround themselves with a reg,on called the ergasphere in which space and ■me are so strongly twisted that all particles, photons, and even local Lorentz frames are forced to rotate about the hole. Lorentz ppn A ' fTu 6iStanCe ! fr ° m a b ' aCk h ° !e ' ' tS 8 ravl ^tional field behaves as if generated by an ordinary star of mass M, obeying the newtonian inverse- square law. Close to a black hole, the gravitational field is far stronger than mas" blar k P h1 1Cted ^ ^^^ theorv - A ™" being sucked into a solar- mass black hole would be npped to shreds by the differential force of gravitv across his body long before he reached the horizon Once the hole's horizon has formed we cannot receive any information lap" r dial dT^ ate °u ^ ?*«*"* "" ^ C ^°™ ° f ^ col ume andTnr 6 > A ff' ^^ that Ae Star 1S Crushed *> «"> vol- ume and mfmite dens.ty at the center of the black hole, forming there a point of infinite gravitational force called. a singularity. For a collapsing star of a few olar masses, the final stellar death throes would be over in a few hundred la°ss^ t 3 f C ° nd ( f S meaSUr6d l0Cally) - Quantum effects ' omi «ed in the classical theory of general relativity, could possibly halt the stellar collapse at onsH T 8 7 t K " Sity P ~ C '' hGi ~ 5 X 10 " S m/Cm: ' Where * > s Pick's prevent thTfc ^"Tu, ?*? ," ° f sin 8 uIa » tle ^ but such effects could not prevent the formation of black holes. , nn C ";, ent th J m ! eS u ° f Stellar evo,utl0n suggest that there could be as many as 100 million black holes in our galaxy. The search for these black holes is not easy We could never detect a tiny black speck a few miles across against the mght sky. Instead we must search for signs of black holes interacting with their astrophysical neighborhoods. Most vulnerable to discovery would be a black hole orbiting a normal star in a binary system. Using Kepler's laws, anal- ysis of the magnitude and period of the doppler shift of the visible normal star a lows us to compute whether the unseen companion is massive enough to be a black hole In addition we might observe intense,, flickering x-rays produced when gas from the normal star is sucked toward the black hole "and heated to a billion degrees in its spiraling path to destruction in the hole. Such are the o a nTk , k bmary X " aV S ° UrCe CygnuS X- 1 ' an e * ceil <^ «mdidate for a black hole about 8000 light-years from earth in the constellation Cvgnus Black holes are a fundamental phenomenon of nature. Space may be littered with black holes tempting us with their secrets of time and space behind a cloak of impenetrable darkness, a challenge and a prize for the persistence of astronomers and physicists. Name: Black Hole Test Per Part A: True/False For each of the following, circle the T if the statement is true. If the statement is false, circle the F, and rewrite as a true statement on the line beneath. T F 1. Light can escape from a black hole. T F 2. Black holes were theoretically discovered by Oppenheimer and Snyder in 1939. T F 3. The theory used to predict black holes is Einstein's Theory of General Relativity. T F 4. Clocks go faster near a black hole. T F 5. Distances are shortened near a black hole. T F 6. A black hole can be characterized by only three quantities: mass, angular momentum and net electrical charge. T F 7. By looking at a black hole you can tell how it was formed. T F 8. A black hole the mass of the sun would have a Schwarzschild radius of 2.95 metres. Part B : Matching In the space to the left, write the letter of the description that best fits each term. Term : .9. Schwarzschild hole .10. A Black Hole .11. Ergosphere .12. Kerr hole .13. Horizon .14. Light .15. Singularity .16. Space Description : A. an open boundary within which the escape velocity exceeds the velocity of light. B. venturing too near a black hole can be bent into circular orbit. C. point of infinite density at the centre of a black hole D. may be littered with black holes. E. are formed when massive stars undergo total gravitational collapse. F. region around a stationary black hole. G. a non-rotating black hole. H. a rotating black hole. I. a closed boundary within which the escape velocity exceeds the velocity of light. J. region around a spinning black hole. Part C : Multiple Choice In each of the following questions, put your answer in the space provided. 17. The theory that implicitly predicts the existence of black holes is? 17, A. the special theory of relativity B. the general theory of relativity C. the theory of singularities D. the theory of stellar evolution 18. Stars balance themselves against their collapse? 18 A. by gravitational force C. B. by spinning very fast D. by thermal pressure by nuclear forces 19. A star will collapse into a black hole when 19. A. its mass is less than three solar masses B. it is spinning at three times that of the sun C. its mass is greater than three solar masses D. it starts to become a white dwarf or neutron star 20. Once the black hole's horizon has been formed 20. A. we cannot receive any information from within the black hole B. its mass is crushed to zero volume at the centre C. its mass is crushed to infinite density at the centre D. all of the above 21. To detect a black hole 21. A. we can use a telescope to see then directly B. we cannot see them with a telescope C. we need to watch for the Doppler shift of a black hole D. we need to watch for the Doppler shift of a companion star 22. The interior of a black hole 22. A. is causally connected with the outside world B. is causually disconnected with the outside world C. is visible to a distant observer using an optical telescope D. is visible to a distant observer using a radio telescope 23. At great distance from a black hole 23. A. space is stretched or warped B. time slows down C. space behaves as if generated by a normal star D. we don't see or feel the black hole 24. A man falling into a black hole 24. E. would not feel a thing F. would find his watch going slower G. would be ripped to shreds H. would feel like he was be compressed Part D : Short Answers 25. Using the principle of conservation of energy, prove that the radius of a black hole of mass M is the Schwarzschild Radius Rs = 2 GM/c 2 . (4 marks) 26. Calculate the Schwarzschild Radius of the earth in metres. (4 marks) mass of the earth = 5.98 x 10 24 kg c = 3x10 8 m/s G = 6.67x1CT 11 Nm 2 /kg 2 27. Describe how you might use a telescope to find a black hole (100 words or less) (4 marks) . 28. In a gravitational field the frequency of light changes as light climbs out of the gravitational well. Starting with the expression for the change in gravitational energy U = mgz where z is the height above the star and using the Planck expression E = h v, where v is the frequency of the light find an expression for the change in frequency of the emitted light. (6 marks) 29. Estimate the fractional change in frequency for light that climbs 10 metres from the surface of the earth. (4 marks) 30. Given the expression for the fractional change in frequency for light that climbs out of a gravitational well, speculate what happens to the light from a star that begins to collapse into a black hole. ( 100 words or less) (4 marks)